Convex Optimization - Cones

A non empty set C in $\mathbb{R}^n$ is said to be cone with vertex 0 if $x \in C\Rightarrow \lambda x \in C \forall \lambda \geq 0$.

A set C is a convex cone if it convex as well as cone.

For example, $y=\left | x \right |$ is not a convex cone because it is not convex.

But, $y \geq \left | x \right |$ is a convex cone because it is convex as well as cone.

Note − A cone C is convex if and only if for any $x,y \in C, x+y \in C$.

Proof

Since C is cone, for $x,y \in C \Rightarrow \lambda x \in C$ and $\mu y \in C \:\forall \:\lambda, \mu \geq 0$

C is convex if $\lambda x + \left ( 1-\lambda \right )y \in C \: \forall \:\lambda \in \left ( 0, 1 \right )$

Since C is cone, $\lambda x \in C$ and $\left ( 1-\lambda \right )y \in C \Leftrightarrow x,y \in C$

Thus C is convex if $x+y \in C$

In general, if $x_1,x_2 \in C$, then, $\lambda_1x_1+\lambda_2x_2 \in C, \forall \lambda_1,\lambda_2 \geq 0$

Examples

• The conic combination of infinite set of vectors in $\mathbb{R}^n$ is a convex cone.

• Any empty set is a convex cone.

• Any linear function is a convex cone.

• Since a hyperplane is linear, it is also a convex cone.

• Closed half spaces are also convex cones.

Note − The intersection of two convex cones is a convex cone but their union may or may not be a convex cone.